PLASMA HEATING AND THE IGNITION PROBLEM

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PLASMA HEATING AND THE IGNITION PROBLEM

E. Canobbio

To cite this version:

E. Canobbio. PLASMA HEATING AND THE IGNITION PROBLEM. Journal de Physique Collo- ques, 1973, 34 (C2), pp.C2-41-C2-47. �10.1051/jphyscol:1973210�. �jpa-00215257�

JOURNAL DE PHYSIQUE Colloque C2, suppliment a u no 11-12, Tome 34, Novembre-De'cembre 1973, page C2-41

PLASMA HEATING AND THE IGNITION PROBLEM

E. CANOBBIO

Association EURATOM-CEA, DBpartement de Physique du Plasma et de la Fusion ContrBlee Centre &Etudes NuclCaires, BP 85, 38041 Grenoble, France

R6sum6. - On ktudie les differentes methodes de chauffage du plasma qui peuvent btre utilisees afin de surmonter les limitations du chauffage ohmique dans la rkalisation des conditions d'ignition d'un plasma toroldal a beta faible.

On discute tout d'abord 1'6quation du bilan global de puissance pour un plasma <( nCoclassique >>

en l'absence et en prksence d'une source de puissance suppl6mentaire. On examine ensuite a) la compression adiabatique, 6) I'injection de faisceaux d'atomes neutres, et c) les chauffages HF rela- tivement lents. On ne traite pas les processus plus rapides tels que les ondes de choc, la turbulence, les champs HF de grande amplitude, les faisceaux d'6lectroris relativistes et les chauffages par laser, qui sont kvidemment inutilisables dans des configurations torofdales a beta faible de taille thermonucleaire.

Les methodes de chauffage HF les plus intkressantes sont, par ordre croissant de frkquence : le Pompage Magnetique par Temps de Transit, 1'Absorption Cyclotron Ionique (dans deux ver- sions diffkrentes) et le Chauffage a la Resonance Hybride Infkrieure.

On donne une explication simple du mecanisme physique qui est a la base dechacune des mkthodes et on signale les aspects qui restent a kclaircir. On indique kgalement les gammes de parametres dans lesquelles des rksultats positifs ont dkja Ct6 obtenus ou peuvent btre atteints par des ameliora- tions de la technologie.

Abstract. - The paper deals with the various plasma heating methods which can be used to overcome the limitations of Ohmic heating in order to achieve ignition in low-8 toroidal systems.

The overall power balance equation of a <( neoclassical >> plasma is discussed without and with the addition of a given amount of supplementary power. Then attention is directed to a) Adiabatic Compression, b) Neutral Beam Injection, and c) slow HF-plasma interactions, leaving aside those fast processes like Shocks, Turbulence, Intense HF-Fields, Relativistic Electron Beams, Lasers, which are obviously impracticable in large energetic low-8 toroidal systems. The most interesting HF Heatings are, in order of increasing frequency : Transit Time Magnetic Pumping, Ion Cyclotron Absorption (in 2 versions) and Heating at the Lower Hybrid Resonance. For each method we briefly explain the basic physical mechanisms involved and point out the important aspects which are not yet well understood. At the same time, we indicate the range of parameters in which these various methods have already been proved to work or are expected to be successful after some technological improvement.

1. Ohmic heating and the ignition problem. -

A feasibility test of the ignition process in low$

toroidal devices (Tokamaks and Stellarators) requires a substantial improvement of the plasma parameters compared to those of present day facilities. This is shown in the following table, where Ti is the ion temperature, n the plasma density, and z, the energy confinement time.

Present Devices Ignition Experiment

- -

Ti several 100 eV E 10 keV

~ Z E cz 10'' cm-3 s 2 1014 cm-3 s Such a progress apparently depends on :

a) increasing the toroidal current I, from a few 100 kA to a few MA (in Tokamaks, I, is the plasma current, in Stellarators, I, is an equivalent external current) ;

b) increasing the plasma volume V by a factor o

-- 30 by keeping constant the aspect ratio R/a

-

(R is the major radius, a the minor radius of the plasma). The toroidal magnetic field necessary to have plasma stability can be 5 50 k G also in an Ignition Experiment, but it must be 2 100 k G in a full scaIe reactor ;

c ) applying auxiliary (non-ohmic) heating methods since the power released by classical ohmic heating

~ Q ( w ) la)^ I;(*)/R(~~) K(ev) 312 (1) is limited to a few 100 kW both in present devices and in the much larger ignition experiments.

In order to see 1) whether ohmic heating is by itself sufficient to achieve ignition, and 2) what is the effect of the addition of a given amount of supple- mentary power, P,, we consider the overall power balance equation [I] :

P* + ^Pa + ^{P" = PB,} + P,, + P,, , (2)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973210

C2-42 E. CANOBBIO

where Pa is the thermonuclear power input from alpha-particles

PB, is the Bremsstrahlung loss

PBr n2 V . T~~~~ , (4) P,, is the loss due to ion thermal conduction as given by the cc banana ⁾⁾ model of the neoclassical theory :

(therefore, PB, > P,,, if I, 2 3 MA), and P,, is the loss by synchrotron radiation.

Since

P,, Ji. T: (6)

synchrotron loss turns out to be relatively unimpor- tant unless Te 2 10 keV and n 5 2 x 1013 ~ m - ~ .

If we assume T, = Ti = T, and define the self- sustaining temperature Tc by the equation

we find that Tc depends on I, only and Tc 5 10 keV if I, 2 2 MA. Then, if we define the temperature T, which can be reached by ohmic heating

Considering the T-n plane (keeping I, constant) one could prove that :

a) by straining every assumption to its most optimistic limit (e. g. absence of any impurities) the To-curve is connected with the T,-curve, i. e., ignition by ohmic heating alone is possible, but only at a density value of

-

b) under reasonable assumptions, even modest amounts of added heat (several 100 kW) can introduce impressive changes in Tokamaks and Stellarators of high performance, and can greatly simplify the ignition of a reactor plasma at low density ;

c ) larger amounts of added heat (10-100 MW)

are necessary to make ignition possible at full operat- ing density.

Concerning point a) we should keep in mind, howe- ver, that at a density of 5 3 x lox3 cm-3 and a temperature -- 10 key, the energy confinement time is as large as 10 s or 2 lo4 z,, where z, is the Bohm

diffusion time. This seems to imply extremely small plasma fluctuation levels and/or magnetic field asymmetries. It is worthwhile to mention that in all present devices z, 5 300 z,. Measurements made in Princeton on the FM-1 hard-core device show [2]

that z,

-

above this temperature !

Finally, we notice that there seems t c be little hope to enhance the plasma resistivity in a substantial fraction of the plasma volume over the purely colli- sional value used in eq. (I), for example as a result of current driven micro-instabilities. As a matter of fact, the ratio between the mean drift velocity of the current carrying electrons and the sound speed is proportional to (a &)-I and will be well below 1 for plausible ignition parameters.

The fundamental weakness of ohmic heating in bringing Tokamaks to ignition serves to keep atten- tion directed to :

a) adiabatic compression, b) neutral beam injection, c) slow HF-interactions, as alternative heating mechanisms.

Fast heating processes like shocks, turbulence, intense HF-fields, which are at present probably the cheapest and most flexible ways to produce kilojoules of plasma energy, imply, when approach- ing reactor parameters, exceedingly expensive energy storage and enormous technical difficulties.

On the other hand, Relativistic Electron Beams (which can deliver as much as lo5 J to a target) cannot be introduced in large toroidal traps for they are just designed to be successful1 in preventing charged particle motion across the magnetic surfaces.

2. Adiabatic Toroidal Compression (ATC). - Three magnetic field components are needed to main- tain equilibrium and stability of a Tokamak : the toroidal component B,, the poloidal component B,, and the vertical field B,. It has been shown [3]

that the best way to compress a torus is to reduce R by varying B, only. This must be made slowly compa- red with the collision time zCol1, but fast compared with z, :

ZE > ZATC > . (8)

An ATC is a reversible transformation whose scaling laws follow immediately from the assumption that the plasma is an ideal gas

= Cte , (9) with infinite electric conductivity, which implies magnetic flux conservation

nu2 B, = Cte , 2 nRaB, = Cte . (1 9)

PLASMA HEATING AND THE IGNITION PROBLEM ^C2-43 Then,

I, ^-+ R - I , etc ... (1 1)

Notice that P, + Cte.

The Princeton ATC-device [4] has proved the efficiency of this method by compressing a plasma with

to a final position R2 -- 38 cm, a , = 11 cm. The results are

which agree with the ATC laws (1 I), while the electron emperature is only T,,

-

From laws (11) and eq. (I), (4) and (5) one can show that a fourfold improvement of nT by ATC as compared to the nT attainable by simple ohmic heating directly at the same R2 position, can only be obtained if PB, > P,,. Otherwise an adiabatic compression is of minor advantage.

An ATC ignition experiment would require R , 5.5 m, R,/R2 = 3, B,, ^1.30 kG. It would be say 3 times more expensive than a conventional device !

Finally we must keep in mind that the compressed stage of an ATC, could never correspond to the runn- ing parameters of a reactor since the average beta value inside the blanket would be too small. However, this process could compress up to ignition a small volume of plasma ; then again alpha-particles heating could be used to bring freshly injected gas to ignition as the full volume which is available for the plasma is slowly filled.

3 . Neutral Beam Injection (NBI). - Fast neutral

atoms penetrate across magnetic surfaces, can ionize well inside a dense plasma, and then heat the plasma quite uniformly by ion-ion and ion-electron collisions.

Small currents of fast neutrals produce at high tem- peratures more collisional heating than large currents of thermal electrons (responsible for the ohmic heating). Thus the equilibrium of the plasma is much less perturbed by NBI than by ohmic heating.

If the kinetic energy of the atoms

ionization takes place mainly on the plasma ions, and the ionization rate v,

-

-

^-

and

E,

-

-- 50-150 keV for an Ignition experiment ,

E 1-5 MeV for a full density reactor .

Since in all cases E,, % T, NBI is really an heating process rather than a density source.

By comparing the ion slowing down forces due to the ions and the electrons of the plasma, one can prove that ions are heated primarily if Eo 5 40 T,, which is the case in present day devices but not in reactor plasmas. There the plasma ions are heated by electron-ion collisions in an equipartition time.

The equivalent neutral beam current INB required to double Ti in present generation devices turns out to be roughly the same as that required to ignite a reactor : for instance 500 kW at 50 keV gives INB = 10 A, just as 30 MW at 3 MeV.

The following data are typical for the present effort in NBI in Tokamaks.

3.1 CLEO-TOKAMAK (UK) : I,

-

P,

-

3.2 ORMAK (USA) : I, 150 kA. - Two 125 kW, 30 keV injectors tangent to the magnetic axis are now assembled, one injecting parallel to I,, the other antiparallel to I,. Ti is expected to grow from 0,4 keV t o 1 keV in 60 ms. Four such sources are planned for the next future.

3.3 TFR (France) : I, 5 ^{400 kA. -} In order to enhance Ti from say 2 keV to 5 keV in 200 ms, a number of Oak-Ridge units giving 11 A of 25 kW (300 kW) neutrals should be used. Lack of accessibi- lity for tangent injection imposes injecting almost perpendicular to the magnetic axis.

3 . 4 An ignition experiment would require inject- ing 2 MW of neutrals from (7-14) units giving (280- 140) kW at 50-100 keV during

-

Areas for future work are : large-surface plasma sources, cooling techniques, larger electrodes, under- standing space charge neutralization, higher energies, D- beams.

Finally, we mention a number of physical effects associated with the interaction of neutral beams with a toroidal plasma. They are predicted by theory and deserve a detailed experimental study :

e microinstabilities induced by the anisotropy of the particle velocity distribution function [5] ;

electric fields stemming from charge separation due to the large radial excursions of the injected ions [6] ;

nonambipolar radial flux of the plasma during injection [6] !

poloidal and toroidal rotations of the whole plasma torus [6].

C2-44 E. CANOBBIO Injecting pairs of tangent beams, parallel and anti-

parallel to the plasma current I,, should be the NBI scheme which minimizes the plasma perturba- tion.

4. High-frequency heating. - MW-CW rf-power is aready available from single units below say 3 GHz.

In this frequency range there is a large variety both of waves which can exist in a magnetized plasma

and of methods to excite them for the purpose of plasma heating. The useful frequency and wave length ranges, the corresponding heating process, and the available CW-power from a single unit are given in the following table, where vti,(,) is the ion (electron) thermal speed, the subscript refers to the lines of force, vCi,(,, is the gyrofrequency, vPi,(,! is the Langmuir frequency, and v,, is the lower hybr~d frequency (see below).

Frequency Wave length

v I Heating frequency CW-rf power

- - - -

150 kHz 2 k m ion TTMP 2 5 M W

v N v t i / I I l

10 MHz 30 m Electron TTMP > 1 MW

v '"te/Iil

50-150 MHz 6-2 m ICH

v = nvci n = 1,2 1 MW

1-3 GHz 30-10 cm LHRH 0.5 MW

V ^N VLH

3 100 GHz < 3 m m Electron heating 5 3 k W

v = vpe ; v N vce.n Above v

-

steeply as

-

The most convenient value of the heating time zH = T/(dT/dt) satisfies the two conditions

the first of which is necessary in order to preserve thermal equilibrium. In present day devices z, 5 10 ms, while with reactor parameters z, 5 1 s.

The most appropriate processes which increase the internal energy of the plasma are the linear wave- particle resonant interactions occurring when

v - nv, = vll/;lll , ^{n = 0, ) 1, f 2,} ... (13) since they are efficient even at low EM-field ampli- tudes. In order to have power transfer from the wave to the plasma, there must be more particles with vll < vRes then particles with vll > vRes. There- fore :

af/avll ^(ull = D R ~ ~ ) < (I4) If n = 0, eq. (13) corresponds to Cerenkov absorption, if I n I ⁼ 1 to cyclotron absorption, and if I n I > 1 to harmonic cyclotron absorption. In the two last cases eq. (13) cannot be satisfied on the whole volume of the torus, but only within a shell of thickness Ax with

When n = 0, the most appropriate wave frequency for ion heating is

v N IJti/Ill < vti/2 ^{7CfLi =} vci (17) where pLi is the ion Larmor radius.

Moreover, vti < v, 6 c (v, is the Alfv6n speed) so that the dispersion relation of the EM-waves reduces to

K: + ~i ⁼ K i .O(V~/V;) ^N 0 , (18) which corresponds essentially to an evanescent vacuum- wave (mainly magnetic). The radial attenuation is, however, irrelevant if 1 2 .na/lll I 5 +. Since there is no wave propagation along the torus, we must put an array of 2 7~R/(;l~~/2) rf-coils around the torus in order to have the EM-pump everywhere.

One can use 1) conventional meridian coils with j,, I Bt (Fig. la) to create a compression B, JI B,, or 2) toroidal coils with B,, 11 j, (Fig. lb) to create a vertical El-field which is linked to a radial B,-field.

The latter is the torsional TTMP first discussed in reference [7].

Finally there is within the plasma an electrostatic field E l l ,

-

From single particle point of view, power absorp- tion is determined by the energy equation in the drift approximation

around the cylindrical surface which is defined by d 1 ^{,,2 =} aBlll,aH fT, ⁺ eVD E , , , (19)

d t -- - '2

v = nv,(R) . (16)

where p is the magnetic moment and VD is the (ver- 5. Transit Time Magnetic Pumping (TTMP). - tical) toroidal drift velocity. The first term on the

PLASMA HEATING AND THE IGNITION PROBLEM C2-45

RHS of eq. (19) - the betatron term - gives the compressional TTMP, the second term is the elec- trostatic (Landau) contribution, the last term is

"esponsible for the Torsional TTMP in toroidal geometry.

From the macroscopic point of view, power absorp- tion is given by the equation for the internal energy [8] :

d l ²

- 5 nmv ^N - p grad v r ~ T / T , , (20) where p is the pressure tensor and v the fluid velocity.

If we plot the heating rate 7;' versus the collision frequency vcOll, keeping constant the modulation rate b = 1 BI Ill Bt I ^{9 (21)}

we find that there are three different regimes (Fig. 2).

Collisions dominate in (A) :

vCol1 > o and zG1 ~ 1 . m2 b2/voll . ⁽²²⁾

In range (B) :

o > vcoll > vor N mb3I2 and ^{.7', cl} o b 2 (23) collisions are sufficient to prevent local distortions

of the distribution function of resonant particles, but are few enough to make the collisionless linear theory to be valid.

In range (C) :

0 < vCol1 < v,, and ^{T ,} ' ^{N v,,,,} . Jb (24) collisions are insufficient to inhibit the nonlinear distortions of the distribution function but are essential in determining power absorption.

In range (B) a modulation b -- a few lo-' is neces- sary to double Ti in present day experiments, while b 5 l o w 3 is enough to achieve ignition in large dense devices. In both cases the electric field at the meridian coils (radius R,) turns out to be 100-150 V/cm :

(TD is the deuterium temperature). A similar analysis in the case of electron TTMP would give I0 times larger E, for the same z, [9].

In moderate scale devices where we want to put into the plasma only 100-200 kW additional power, the power of the TTMP oscillator must nevertheless be as large as

-

A recent experiment on Proto-Cleo (Culham Laboratory) [lo], as well as a theoretical paper [Ill, indicates that in order to prevent enhanced plasma losses during the heating phase it is necessary to put electrostatic screening around the rf-coils.

To prevent shielding of the electro-magnetic fields, the rf-coils should be put inside the vacuum chamber.

However, arcing and surface problems impose t o insert between coils and plasma an rf-transparent liner which can be either an insulator or an all-metal structure cut parallel to B, [12].

The first ad hoc TTMP experiment will be performed in 1974 on the Petula-Tokamak of the Grenoble Laboratory.

6. Ion Cyclotron Heating (ICH). - The ion cyclo- tron wave (B, I BJ can propagate around a torus if v < vci and m = 1, where rn is the poloidal angular wave number. Then, the wave can be excited in a small sector of the torus by using one single rf- coil structure. An excellent coupling to the plasma has been realized in Kharkov, by using the coil represented on figure 3a [13].

A satisfactory result has also been obtained on the ST-Tokamak in Princeton with a single rudimentary half-turn coil figure 3b [14]. In both cases heating of moderately dense ( 5 lot3 ~ m - ~ ) plasmas was achieved., However, ICH becomes inefficient in larger and denser plasmas since the electric field which rotates in the same sense as the ions is strongly screened by the plasma currents if

K ; c2 5 : ~ ; ~ ^{or A,,} 2 2 x lo8/& (cm) . (26)

4

C2-46 E. CANOBBIO

On the other hand, All cannot be made arbitrarily small because 6 (the rf-coil-plasma distance) cannot be made much larger than A1,/2 n, otherwise the rf- power incident on the plasma would be reduced by a factor

The efficiency of ICH can be increased substantially in a two-ion species dense plasma if

since then the electric field rotating with the reso- nant ions would penetrate easily into the plasma.

Now the E, value at the coils becomes comparable with the E, value required for TTMP, for the same 2,-value. This method has not yet been tested expe- rimentally.

7. Harmonic ion cyclotron heating. - The fast magnetoacoustic wave (B, // B,) propagates around the torus if

The condition to have radial plasma eigenmodes is

Taking account of the ion Larmor motion, the field of an EM-wave as seen by the ion takes the form

ei(2nvt -KIX) , - e i ( 2 n v t - K l p ~ cos 2nvCit) N

Then, if v = 2 vci, the ions experience a wave with v = vci and K ^N 2 mpi/c 4;. The wave-particle interaction can be quite strong at high plasma den- sity, and heating comparable with that of a two- ion plasma can be produced.

Incouraging preliminary results at low rf-power level have been obtained on the ST-Tokamak, where a MW-experiment is . planned for Spring 1974.

We remark finally, that the ions which are prefe- rentially heated at v = 2 vci have small parallel energy and mainly gain perpendicular energy. Therefore, they tend unfortunately to escape from the confi- guration via banana orbits.

8. Lower-Hybrid Resonance Heating (LHRH). -

If I n I % 1 Harmonic Cyclotron Heating is inefficient unless the EM-field be strongly amplified within the plasma. Amplification of the radial component of the electric field does occur in a plasma near the surface where K,/KI, -+ co

The EM-power penetrates from the outside up to the resonance surface if the parallel wave number satisfies the condition [15]

It is therefore necessary to place around the plasma column (if it is thicker than Aovci/vpi) a slow wave structure at a distance

With thermonuclear parameters, 6 should be less than a few cm ! From a large number of experiments all over the world, there is now increasing evidence of good coupling efficiency and heating at LHR.

Heating in such experiments is probably due to some non-linear effect occurring when the field inside the plasma is amplified above some critical value, rather than to the linear interaction at v -- nvCi (probably a parametric instability develops, wherein the pump wave decays into 2 other waves). A 100 kW-LHR Heating Experiment will be performed (1975) on the WEGA-Stellarator of the Grenoble Laboratory.

PLASMA HEATING AND THE IGNITION PROBLEM

References

A) General References on Plasma Heating and the Ignition Problem :

CANOBBIO, E., (( Review of plasma heating methods in relation to the reactor ignition problem D, in Proceedings of the International School on Fusion Reactor Technology, Erice, September 1972. EUR-4999e.

CANOBBIO, E., (( Report on the Varenna Symposium 1972 n, Nuclear Fusion 13 (1973) 11 1.

ADAM, J., GIRARD, J. P., SAMAIN, A., ((Rapport du Groupe d'Etude du Chaufage d'un plasma confine' dans une configuration magne'tique de type tokamak x, Rapport EUR-CEA-FC-5179 (1971).

B) Special Topics :

[I] SWEETMAN, D. R., Nuclear Fusion 13 (1973) 157.

[2] YOSHIKAWA, S., IZI International Symposium on Toroidal Plasma Confinement Garching, 26-30 March, 1973, paper C 2.

[3] FURTH, H. P. and YOSHIKAWA, S., Phys. Fluids 13 (1970) 2593.

[4] BOL, K. et al., Phys. Rev. Lett. 29 (1972) 1495.

[5] CORDEY, J. G., HOUGHTON, M. J., Nuclear Fusion 13 (1973) 215.

[6] CANOBBIO, E. and DE BARBIERI, O., VI European Confe- rence on Controlled Fusion and Plasma Physics (Mos- cow 1973) Vol. 1.

[7] KOECHLIN, F. and SAMAIN, A., Phys. Rev. Lett. 26 (1971) 490.

[8] BRAMBILLA, M., Plasma Physics 1973.

[9] CANOBBIO, E., Symposium on Plasma Heating and Injec- tion, Varenna (1972) 14.

[lo] MILLAR, W. et al., ZII International Symposium on Toroidal Plasma Confinement, Garching, 26-30 March 1973, paper E-6.

[11] BRAMBILLA, M., 111 International Symposium on Toroidal Plasma Confinement, Garching, 26-30 March 1973, paper E-7.

[12] CONSOLI, T., Symposium on Plasma Heating and Injection, Varenna (1972) 61.

[13] OVCHINNIKOV, S. S. et al., Nuclear Fusion (Supplement 1972) 347.

[14] HOOKE, W., Symposium on Plasma Heating and Injection, Varenna (1972) 73.

[I51 GLAGOLEV, V. M., Plasma Physics 14 (1972) 301, 315.